Theorem pm5.18dc 777

Description: Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider -. ( ph <-> -. ps ) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)

Assertion

Ref	Expression
pm5.18dc	|- (DECID ph -> (DECID ps -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )

Proof of Theorem pm5.18dc

Step	Hyp	Ref	Expression
1		df-dc 743	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		pm5.501 233	. . . . . . . 8 |- ( ph -> ( -. ps <-> ( ph <-> -. ps ) ) )
3	2	a1d 22	. . . . . . 7 |- ( ph -> (DECID ps -> ( -. ps <-> ( ph <-> -. ps ) ) ) )
4	3	con1biddc 770	. . . . . 6 |- ( ph -> (DECID ps -> ( -. ( ph <-> -. ps ) <-> ps ) ) )
5	4	imp 115	. . . . 5 |- ( ( ph /\ DECID ps ) -> ( -. ( ph <-> -. ps ) <-> ps ) )
6		pm5.501 233	. . . . . 6 |- ( ph -> ( ps <-> ( ph <-> ps ) ) )
7	6	adantr 261	. . . . 5 |- ( ( ph /\ DECID ps ) -> ( ps <-> ( ph <-> ps ) ) )
8	5, 7	bitr2d 178	. . . 4 |- ( ( ph /\ DECID ps ) -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) )
9	8	ex 108	. . 3 |- ( ph -> (DECID ps -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )
10		dcn 746	. . . . . . 7 |- (DECID ps -> DECID -. ps )
11		nbn2 613	. . . . . . . . 9 |- ( -. ph -> ( -. -. ps <-> ( ph <-> -. ps ) ) )
12	11	a1d 22	. . . . . . . 8 |- ( -. ph -> (DECID -. ps -> ( -. -. ps <-> ( ph <-> -. ps ) ) ) )
13	12	con1biddc 770	. . . . . . 7 |- ( -. ph -> (DECID -. ps -> ( -. ( ph <-> -. ps ) <-> -. ps ) ) )
14	10, 13	syl5 28	. . . . . 6 |- ( -. ph -> (DECID ps -> ( -. ( ph <-> -. ps ) <-> -. ps ) ) )
15	14	imp 115	. . . . 5 |- ( ( -. ph /\ DECID ps ) -> ( -. ( ph <-> -. ps ) <-> -. ps ) )
16		nbn2 613	. . . . . 6 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )
17	16	adantr 261	. . . . 5 |- ( ( -. ph /\ DECID ps ) -> ( -. ps <-> ( ph <-> ps ) ) )
18	15, 17	bitr2d 178	. . . 4 |- ( ( -. ph /\ DECID ps ) -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) )
19	18	ex 108	. . 3 |- ( -. ph -> (DECID ps -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )
20	9, 19	jaoi 636	. 2 |- ( ( ph \/ -. ph ) -> (DECID ps -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )
21	1, 20	sylbi 114	1 |- (DECID ph -> (DECID ps -> ( ( ph <-> ps ) <-> -. ( ph <-> -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  xor3dc  1278  dfbi3dc  1288